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From Adventures in Stochastic Process - Sidney Resnick

Firstly, I believe there is an error with the function G on the problem. It should be: $ G(x) = \sum_{k = 0}^{\infty} P[N=k]\cdot F^{k*}(x)$, where $F^{k*}(x) = P[X_1+...+X_k<x]$. Because we don't know the distribution of N. Now, this is how I find the Laplace Transform. $$ \hat{G}=\int_0^\infty e^{-\lambda x}[\sum_{k=0}^\infty P[N=k]F^{k*}(x)]dx \\ =\sum_{k=0}^\infty P[N=k]\int_0^\infty e^{-\lambda x}F^{k*}(x)dx \\ =\sum_{k=0}^\infty P[N=k][\hat{F}(\lambda)]^k $$ where $\hat F$ is defined as the Laplace Transform of function $F$.

Is my answer correct? I'm not sure with the second equality. Thank you.

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Since your integrand is non-negative, the interchange of integral and sum in the first step is justified by Fubini-Tonelli.

$F^{*k}$ is the $k$-fold convolution of $F$ (convolution in the sense of probability distributions).

Thus, the next equality is also correct, since the Laplace transform of a convolution of measures (a probability distribution of a real-valued random variable is a Borel measure on $\mathbb{R}$) the product of the Laplace transforms, i.e.

$$\int_0^\infty e^{-\lambda x} F^{*k}(x)dx=\widehat{F^{*k}}(\lambda)=\widehat{F}(\lambda)^k$$

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